If your ability score maximum is raised above 20, you can use an ASI to go above 20
Most of this answer is copied from my answer to the related question "Can the Bag of Beans increase an ability score above 20?". I've trimmed my answer below for length and relevance, so see the linked question for a deeper analysis.
Characters normally can't have an ability score above 20...
The general rules for ability score maximums are set forth in the "Ability Scores and Modifiers" section of chapter 7 of the PHB and basic rules (emphasis mine):
A score of 10 or 11 is the normal human average, but adventurers and many monsters are a cut above average in most abilities. A score of 18 is the highest that a person usually reaches. Adventurers can have scores as high as 20, and monsters and divine beings can have scores as high as 30.
This paragraph tells you a number of things about ability scores.
- First, 10-11 is the "average" score for humans in general. This
corresponds to a modifier of +0, and the commoner NPC statblock
seems to match this as well, having a score of 10 in all six
abilities. (Both of these facts also seem to fit the unstated
expectation that the rules are written in a way that treats humans as
the "default", common in the Material Plane of every official
setting, at least in 5e.)
- Secondly, a score of 18 is the highest that a person "usually"
achieves. It's not clear exactly what this is referring to or what
the purpose of this sentence is. If I had to guess, it's meant to to
the fact that 18 is the highest possible ability score value before
racial modifiers for a level 1 character (assuming the default method
of determining ability scores: roll 4d6 and drop the lowest die,
- The final sentence says that adventurers' ability scores can be as
high as 20, and that monsters (and "divine beings", possibly
referring to powerful NPCs that might not be considered "monsters")
can have an ability score as high as 30.
In my opinion, this last sentence (bolded in the quote above) is a clear statement of the general rule for PCs: Player characters have an ability score maximum of 20. Barring specific things that overrule this statement, these do describe the maximums for ability scores for adventurers and monsters/NPCs respectively.
The description of the additional Honor ability score option also alludes to the existing limit on player characters' ability scores and extends it to the new score as well (DMG, p. 265; emphasis mine):
Unlike other abilities, Honor can’t be raised with normal ability score increases. Instead, [the DM] can award increases to Honor—or impose reductions—based on a character’s actions. [...] As with other ability scores, a character’s Honor can’t exceed 20 or fall below 1.
This clearly implies that the regular 6 ability scores can't normally exceed 20 or fall below 1 either.
Rules designer Jeremy Crawford also noted in an unofficial tweet from September 2017:
Ability scores go no higher than 30. For regular folk, they go no higher than 20.
In essence, unless specified otherwise (by a class feature, magic item, or something else), the ability score maximum for a player character is 20.
...But there are specific exceptions to the general rule
You may have noticed my disclaimer earlier about "specific things that overrule this statement". The reason I note this, even though it's not mentioned in the quoted rule, is because of one of the fundamental rules of D&D - Specific Beats General:
That said, many racial traits, class features, spells, magic items, monster abilities, and other game elements break the general rules in some way, creating an exception to how the rest of the game works. Remember this: If a specific rule contradicts a general rule, the specific rule wins.
There are a number of things in the game that don't ignore the general rule on a character's ability score maximum, and just a few things that do.
The general rule applies in most cases, including for ASIs
Most things that increase an ability score are limited by the character's ability score maximum. For instance, the Ability Score Improvement class feature, the ioun stone of agility, and all the "half-feats" (which increase an ability score and also grant another benefit) all state that the ability score increase they grant can only bring the character to a maximum of 20.
In essence, they're reiterating the general rule on a character's ability score maximum, not making up a specific rule where no general rule exists. It's a redundant clarification, but one that is helpful to make when there are other items that do ignore or generally increase this maximum (as listed below).
Some stuff sets an ability score to a fixed value, ignoring your maximum
There are a few items that essentially disregard your normal score for a certain ability (along with your normal maximum for that score), and simply set that ability score to a particular value (unless your base score is already higher).
For instance, the gauntlets of ogre power simply set your Strength score to 19, unless it's already higher. Similarly, the belt of giant strength changes your Strength score to a value between 21 and 29, depending on the variety. (A potion of giant strength has a similar temporary effect.) The headband of intellect sets your Intelligence score to 19, and the amulet of health does the same for Constitution.
All of these items, however, only set the score to that value while attuned (or for 1 hour, for the potion of giant strength. More importantly, they don't interact with your regular ability score maximum in any way; they simply set your ability score to the specified value, unless your "true" ability score (i.e. your score without the temporary boost) is already higher - in which case they have no effect.
Things that specifically increase your ability score by a particular amount (e.g. ASIs, other magic items) only affect your base ability score, not the fixed value assigned to you by one of these items. Crawford confirms this interpretation in a series of unofficial tweets from August 2015. (Multiclassing prerequisites are also meant to be met by your base score tangentially related Q&A: Can multi-classing prerequisite ability score(s) be met through a magic item?)
Some ability score increases have their own limit, solely for that increase
As mentioned, the Star card from the deck of many things increases an ability score by 2; its description says the score can exceed 20, but can't exceed 24. This reiterates the general rule to explain how the item ignores it; if there were no general rule, there'd be no point to saying the increase can bring the score above 20.
Note, however, that as designer Chris Perkins explained in an unofficial tweet in December 2016:
It increases one ability by 2, to a maximum of 24. It does not increase your maximum for that stat otherwise.
In other words, the Star card doesn't actually change your ability score maximum going forwards; it only raises it (if necessary) to 24 for the increase granted by the card itself, and then your ability score maximum starts to apply again. For example, if your Strength score is 18, and the Star card increases it to 20, gaining an Ability Score Improvement feature later can't then increase your Strength further (unless something has actually increased your ability score maximum from 20 to something higher).
One of the benefits of the Book of Exalted Deeds says it increases Wisdom score by 2, to a max of 24. This is identical to the phrasing of Perkins' tweet above, in that it increases your current score (temporarily increasing the maximum solely for that increase, if needed), but doesn't change your ability score maximum otherwise.
(Ostensibly, if the Star card brings your score to 21-24 and that's above your ability score maximum, then your ability score after the card effectively becomes your new maximum. However, I'm not sure how that new maximum would interact with effects that reduce your ability score. That would be better asked as a separate question.)
Very few things permanently increase your overall limit
In contrast, there are only a few items that explicitly increase an ability score maximum in general. For instance, the manual of gainful exercise explicitly says (emphasis mine):
your Strength score increases by 2, as does your maximum for that score.
There are six such items with this sole benefit, one for each ability:
One of the benefits of the orb of the veil (Explorer's Guide to Wildemount, p. 268) is also identical to that of the tome of understanding.
If there were no general "ability score maximum" rule, the latter part of the quoted statement would be entirely meaningless.
In other words: If we assume that only the Ability Score Improvement feature (as well as other such ability score-increasing features or items) is limited to a maximum score of 20, and otherwise anything else can increase your ability score above 20 (as some other answers suggest), then what does it mean to "increase your maximum for that score" by 2? A nonexistent value can't be increased. If there were no general rule, it'd be easier to just leave off that part of the sentence entirely.
Thus, it's clear that characters do have a default ability score maximum of 20 for each ability score. That maximum can be increased or ignored/overridden by certain items if they explicitly say so, but otherwise generally functions as an upper limit on the character's ability score.
Ability scores can never go above 30; there's no mechanics for it
Though the default ability score maximum of 20 for player characters can be modified by more specific things in the game, the listed ability score maximum of 30 for monsters is ironclad; there's no modifiers listed for scores above 30, and no official monster has any ability score above 30. Also, under the rules for creating a monster stat block, the section "Step 5. Ability Scores and Modifiers" reiterates the relevant rule (DMG, p. 274):
A monster can’t have a score lower than 1 or higher than 30 in any ability.
In fact, this extends to PCs as well - even the exceptions to a character's ability score maximum can't take you above 30, because the game does not allow for scores above 30 or below 1.
Can an ASI increase an ability score above 20?
As I've mentioned above, the Ability Score Improvement class feature says (e.g. for barbarians; emphasis mine):
When you reach 4th level, and again at 8th, 12th, 16th, and 19th level, you can increase one ability score of your choice by 2, or you can increase two ability scores of your choice by 1. As normal, you can’t increase an ability score above 20 using this feature.
The phrasing of this feature clearly indicates that it's simply repeating the "default" rule that a character's ability score maximum is 20 unless something specifically overrides it.
The description of the manual of gainful exercise, as quoted in the question and earlier in my answer, says (emphasis mine):
[...] If you spend 48 hours over a period of 6 days or fewer studying the book's contents and practicing its guidelines, your Strength score increases by 2, as does your maximum for that score. [...]
This explicitly increases your ability score maximum from its previous value. Clearly, the specific magic item is more specific than the general rule. As such, even though the ASI feature says that "as normal," you can't increase a score above 20 with that feature, a magic item that explicitly increases your ability score maximum and thereby overrides the default rule does allow you to increase your current score up to that new maximum using an ASI.
Crawford unofficially confirmed this interpretation for a similar interaction in a series of tweets from March 2018:
If I use a Tome or Manual to go from a 19 to a 21 in a stat and increase my maximum, can I use a "half-feat" like Resilient to get to a 22?
If your ability score maximum is raised to 22, that means you can now raise that score to 22.
If you find multiple books or are long lived, can you get this bonus multiple times?
It wouldn't allow a creature to ignore the max cap of 30, however, would it?
The relevant part of the Resilient feat says (PHB, p. 168):
Increase the chosen ability score by 1, to a maximum of 20.
This feat is also referencing the default ability score maximum of 20, but Crawford's response confirms that if a tome/manual increases your ability score maximum in general, then you can increase your score through other means up to that maximum as well. "Half-feats" and ASIs are two of the ways that a character can do so.